homonuclear diatomic MO diagram

The cleanest place to learn molecular orbital theory is a molecule of two identical atoms — H2, N2, O2, F2. Because both atoms are the same, their atomic orbitals start at exactly the same energy, so the diagram is symmetric: the molecular orbitals sit neatly in the middle, equally shared by both atoms. Walking from H2 across to F2, building each diagram and filling it, is the standard tour of what the theory can do.

For the second-row diatomics (Li2 through F2) each atom brings one 2s and three 2p valence orbitals. The 2s pair makes a bonding sigma2s and antibonding sigma-star2s. The 2p orbitals split by how they point: the pair pointing along the bond overlap head-on to give sigma2p and sigma-star2p, while the two pairs pointing sideways overlap to give two degenerate pi2p orbitals and two pi-star2p orbitals. Fill these with valence electrons bottom-up and you can immediately read each molecule's bond order and magnetism. There is one twist: for the lighter members (B2, C2, N2), s-p mixing pushes the pi2p orbitals just below sigma2p, whereas for O2 and F2 the usual order sigma2p below pi2p is restored.

This single family is the theory's greatest showcase. It correctly gives N2 a bond order of three and diamagnetism, F2 a bond order of one, and — the headline result — O2 a bond order of two with two unpaired electrons, making it paramagnetic. No Lewis structure of O2 can show those unpaired electrons; molecular orbital theory predicts them effortlessly. The diatomics also illustrate trends: as bond order rises across the series, bonds get shorter and stronger, exactly as measured.